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I believe that every mathematical "apprentice" more or less once found that some stuff that had been unclear somehow became clear; for example, some books might be unaccessible to you two years ago but totally accessible now. Even having these experiences, from time to time I would still be tempted to think that I may not be able to go further in mathematics when facing those now-unaccessible stuff. Now my question is how you develop a good mindset to turn away from the time-wasting behavior of doubting your own ability?

This question is possibly opinion-based, in all conscience; but it depends on people's flavor.

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    $\begingroup$ According to Paul R. Halmos, Measure theory, preface, "...[the reader] should not be discouraged, if on first reading of section 0, he finds that he does not have the prerequisites for reading the prerequisites." -- the problem you are having has been known to be quite common for a while. $\endgroup$
    – Thomas
    Oct 4 '15 at 12:10
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    $\begingroup$ "What one fool can do, so can another." -- Richard Feynman $\endgroup$
    – littleO
    Oct 4 '15 at 12:41
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    $\begingroup$ "Everything is hard before it is easy" - Goethe $\endgroup$ Oct 4 '15 at 12:57
  • $\begingroup$ "You've got to be very careful if you don't know where you are going, because you might not get there." - Yogi Berra $\endgroup$ Oct 4 '15 at 13:12
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    $\begingroup$ Richard Feynman sometimes joked about his own ability,but he got a Nobel prize in physics...... From time to time, take a look at earlier material and think about it again, and recall how it was when you started it. And try some new elementary problems.You may find them refreshing. Even great mathematicians take interest in elementary Q's. An interesting book, which I think is by Felix Klein, is Elementary Mathematics From An Advanced Standpoint. $\endgroup$ Oct 4 '15 at 13:15
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I guess we all encounter some "heavy" material from time to time and feel pressure for not being able to understand everything perfectly. This is perfectly normal, or to rephrase it in the immortal words of von Neumann:

"Young man, in mathematics you don't understand things. You just get used to them."

Personally, as long as I can remember, I've done maths and I've been good at it. I've decided to study mathematics when I was 10, or something like that. I am far from being very talented mathematician or very hard working one. I simply love maths and for that reason alone I've always felt that there is nothing in mathematics that I won't eventually be able to grasp. So, when I feel frustrated about a subject I'm trying to learn, it's only temporary as I'm aware it will become trivial soon enough, just like all the stuff I learned before.

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    $\begingroup$ Boy oh boy do I strongly disagree with that quote.... $\endgroup$
    – Wildcard
    Oct 4 '15 at 12:21
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    $\begingroup$ @Wildcard: I would like to listen if you would like to explain; thanks. $\endgroup$
    – Megadeth
    Oct 4 '15 at 14:04
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    $\begingroup$ @Wildcard, I guess it depends on interpretation, but I don't take it that literally. I feel that it should be seen as no understanding will come before you are well used to notions and manipulations involved, before you have many examples and counterexamples. Only after will you be able to predict behaviors and solve problems (at least conceptually) by glancing at them or with little effort, and you may call that understanding, if you like. $\endgroup$
    – Ennar
    Oct 4 '15 at 21:26
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    $\begingroup$ @Ennar, I actually like your answer, a whole lot. I just disagree with the quote in it. I prefer the line from you: "...there is nothing in mathematics that I won't eventually be able to grasp." I guess you're interpreting the quote to mean roughly the same as your version, but the literal meaning of that quote is quite different. (In other words, I prefer you to von Neumann.) ;) $\endgroup$
    – Wildcard
    Oct 4 '15 at 21:29
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    $\begingroup$ @Wildcard, thank you for your kind words. I just wanted to say how I interpret von Neumann's quote as it defeats the purpose of the rest of my post if understood your way. I'd like that anyone who reads my answer feels positive about mathematics, and not feel depressed if stuck on a problem. $\endgroup$
    – Ennar
    Oct 4 '15 at 21:40
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If it's a book on mathematics, a person wrote it. If a person wrote it, another person (you) can understand it.

At least, you can understand it at least as well as the person who wrote it. In some cases that might end up being "not very well" ;) , but that's when it's time to dive for a more original, more basic textbook...if possible, one from the original inventor of that branch of mathematics.

And yes, this is primarily opinion-based. The generic translation of this question is "How to achieve self confidence?" I have more strongly held opinions about this, but those would be even less appropriate here than this question is.

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    $\begingroup$ Of course, you have to hope that the person who wrote the book wrote it with the intent for someone to understand it! Z quorblax als raimre ipsum. $\endgroup$ Oct 4 '15 at 18:23
  • $\begingroup$ "If a person wrote it, another person (you) can understand it." ... are you a betting man ;) $\endgroup$
    – BAR
    Oct 5 '15 at 1:21
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I am experiencing precisely what you are describing - I am approaching a realm of mathematics new to me, in fact, I did not know it existed until 3 weeks ago.

It will probably sound a bit cliched, but I have found some good effective strategies for developing the positive focused mindset that you describe involve:

  • Starting from an application based context for the mathematical principles, relating it to either real-life applications or being analogous to them.
  • I go with the mindset of starting from the beginning, got to start somewhere.
  • The notion of challenging oneself.
  • Most of all, perseverance and practise.

Remember, 'failing' at understanding something initially just eliminates that approach, the following meme basically says it all:

enter image description here

Image source

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Personally, I think back to times I previously thought this, and subsequently learnt the material anyway. It's especially noticeable in my current project of working through Awodey's Category Theory. Every so often I flick through the future chapters of the book and ask myself, "Can I understand the words on this page yet?" The answer is always no. Yet after I've read through a chapter and done the exercises, and thought around it a bit, I understand its content.

Similarly, I used not to know what a linear map was. I would look at any mathematical Wikipedia page and reel back from it in horror. Now, there's a reasonable chance that I will at least be able to parse a given Wikipedia mathematics page.

Remember the number of times in the past you've thought you couldn't do it, and the number of times in the past you became able to do it after working hard.

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